Philosophy from the mind of a fighter

Drunvalo Melchizedek is Bad at Math

I decided to go through The Ancient Secret of the Flower of Life with a fine-toothed comb in order to determine the veracity of the mathematical claims which Drunvalo Melchizedek makes in his most well known work. This post represents my review of every single mathematical claim which I could find in ASoFoL Volume 1. For a book which is purported to be focused primarily on geometry, I found surprisingly little mathematical information. Out of its 225 pages of material, only 32 pages mentioned any mathematical principles. Nearly 86% of the pages in this book have absolutely nothing to do with mathematics or geometry.

Now, please note that I have restricted myself only to dealing with the mathematical claims in ASoFoL. I have deliberately ignored dealing with any claims relating to anatomy, astronomy, biology, history, physics, or theology. While a great number of these claims are just as problematic as the ones I have listed below, I felt that I needed to restrict myself to a single field of knowledge, in order to avoid writing a document which ends up being longer than the one I am reviewing.

In the following entries, a single letter will precede the page number where the claim can be found. An “A” denotes a claim which I have found to be entirely accurate, while an “I” denotes a claim which is, either in whole or in part, inaccurate. If anyone finds any additional claims in Volume 1 which I ignored, please bring them to my attention.

2. I: pg 45 – Vesica piscis
The description of the vesica piscis, itself, is accurate. However, he implies that the proportion of the major axis of the vesica piscis to the minor axis is the Golden Mean. This is not true. That proportion is to 1.

3. I: pg 46 – “Musical notes are separated by 90 degrees”
How are musical notes separated by 90 degrees? Certainly not in frequency or proportionality, as these are not angular measurements.

4. I: pg 47 – Sine Waves
The diagram containing samples of purported sine waves is inaccurate. The middle diagram is not sinoid, at all, as it’s amplitude and wavelength vary throughout the diagram, while sinoid patterns have uniform amplitude and wavelength. The top diagram is sinoid, but if we assume the leftmost position of the diagram to be x=0, this certainly isn’t representative of . Similarly, with the bottom diagram, while the pattern is sinoid, the pattern is certainly not representative of , presuming the straight axis is the y-axis, and that the leftmost point is x=0.

5. I: pg 49 – “Geometrical progression”
Drunvalo correctly classifies his model of continuing 12-series progression as a “geometrical progression,” but inaccurately remarks that such sequences are only classified this way in harmonics.

6. I: pg 53 – “Number sequences need a minimum of 3 numbers to calculate the entire sequence, except Golden Mean which needs 2”
Drunvalo will repeat this claim on pg 215. However, in neither case does he provide any evidence that this is the case. Nor does it make any sense. For example, the most basic of all number sequences– the natural numbers– only requires a single initial term: 1. Then the next term is 2, then 3, 4, 5, and on, ad infinitum.

7. I: pg 155 – “Genesis Pattern” has minimal lines for a 2D representation of a torus
This statement is simply false. A single line forming an elliptoid is the minimal 2D projection of a torus. Furthermore, the “genesis pattern” is not a projection of a torus, at all.

8. I: pg 156 – Transcendental Numbers
Drunvalo makes several mistakes, all at once, on this page. Firstly, he provides an incorrect– and, frankly, absurd– definition for transcendental numbers. While “transcendental” can mean different things to different people, Drunvalo explicitly states that he is referring the the mathematical understanding of transcendental numbers. In mathematics, a transcendental number has a very specific definition: a number which is not the root of any polynomial with integer coefficients. It has nothing to do with “dimension.”

Furthermore, numbers do not “change” from one dimension to another. Numbers are measurements of dimension. A number which is rational or irrational, algebraic or transcendental will be such regardless of the dimension it measures.

His claim that “you never know what the next digit is going to be” is simply nonsensical. You never know what the next digit of ANY number will be until you calculate it. This property has nothing to do with whether or not a number is transcendental.

He then identifies the Golden Mean as a transcendental number, which is patently incorrect. The Golden Mean is the root of the polynomial .

10. I: pg 161 – Platonic Solids
There are actually an infinite number of unique shapes. What Drunvalo meant to say is that there are only five unique shapes which conform to the properties of regular polyhedra, and these five are affectionately referred to as the Platonic Solids.

11. A: pg 163 – “Pentagonal Dodecahedron”
It is not inaccurate, but why does Drunvalo insist on referring to this solid as a “pentagonal” dodecahedron, as if there were any other possible type of regular dodecahedron? If the solid being discussed is a regular dodecahedron, its faces must be pentagonal. He does not refer to the other shapes in this manner. He never refers to a “tetragonal” cube or a “trigonal” tetra/octa/icosa-hedron.

12. I: pg 164 – Penrose tiling as a shadow projection of Metatron’s Cube
Casting a shadow using a 3-dimensional Metatron’s Cube as the mask does not produce a perfect Penrose tiling, as Drunvalo is attempting to claim. This is empirically testable. While the structure of pentagons highlighted within the Penrose tiling can certainly be cut out and folded to create a dodecahedron, the design was deliberate. It was not simply generated by shadowcasting.

13. I: pg 166 – All you need is a compass and a ruler
If all one needs in order to fully understand and appreciate geometry is a compass and ruler, I would love for Drunvalo to show me how to construct a regular nonagon using only those tools. Surely, if that is all one needs to learn “everything there is to be known about any subject,” then he can show me how to accomplish something within the subject of geometry, itself!

14. I: pg 168 – Rotating a cube 72 degrees describes an icosahedron
Rotating a cube about which axis? According to what pattern? Is the icosahedron described by the position of the cube’s vertices after the rotation? A cube has 8 vertices, while an icosahedron has 12, so which vertices are we counting in constructing the icosahedron? Drunvalo’s assertion, here, is nonsense without further information. It means absolutely nothing.

15. A: pg 180 – Platonic Solids can be inscribed within a Cube or Sphere in perfect symmetry
While this is true enough, the Cube and Sphere inscribe the remaining Platonic solids in very different ways. All five Platonic solids can be inscribed within a Sphere in such a manner that the only intersection between the two figures is at the vertices of the polyhedron, and each vertex lies somewhere on the surface of the sphere. However, the only regular polyhedron which can be contained by a square, in this manner, is the octahedron. The tetrahedron, icosahedron, and dodecahedron cannot be contained within a Cube in this way.

16. I: pg 186 – Two spheres moving through one another’s space until they coincide will, at some point, form a vesica piscis
While Drunvalo’s intended meaning is on the right track, he mis-speaks, here. The spheres never form a vesica piscis, since a vesica piscis is a two-dimensional figure. However, the two-dimensional projection of these spheres will, in fact, form the vesica piscis– provided the projection is taken at the correct angle.

17. I: pg 188 – Four close-packed spheres form a tetrahedron
Four close-packed spheres do NOT form a tetrahedron. Connecting the centers of four close-packed spheres DOES form a tetrahedron.

18. I: pg 195 – Construction from intersecting circle and square
Drunvalo relays a story about a Mason drawing a figure for him, and provides a facsimile of that drawing on the page. The description of the construction of this figure is accurate, but the facsimile then contains some text, reproduced as follows:

Within these seven lines there is one oddity, one ambiguity, and one flat-out mistake. The oddity is the inclusion of D/2 = r. While correct, this equation has nothing to do with the rest of the calculations– Drunvalo does not solve for r, nor does he use r anywhere else in the math. The ambiguity is the line “Ratio = 1.618;” what ratio is he talking about? There is not a single proportional ratio described ANYWHERE in the diagram which is close to 1.618. Finally, the mistake is in “D = 15.2866.” In calculating D, whoever wrote this only used 3 significant digits of π, but gave a result with six digits. Using 6 digits of π, we get D = 15.2788.

19. A: pg 195 – Phi Ratio
Drunvalo’s description of the phi ratio is accurate. It is the proportion of A:B such that A:B::A+B:A, and a decimal approximation of that proportion is 1.618.

20. A: pg 197 – Four tangent circles inscribed in a square and a circle produce equal perimeter and circumference
Drunvalo correctly realizes that the diagram does not produce a square and circle of equal perimeter and circumference. He then posits that one must construct another circle, slightly larger, in order to reach the correct image. While this is correct, he negates to tell us exactly how much larger the outer circle must be. I’ve gone ahead and performed this calculation, for him. The outer circle must have a radius exactly times larger than the inner circle. That is, approximately 0.132 times larger.

21. A: pg 198 – Diagram of the construction of a Golden Section
This diagram accurately describes the construction of a Golden Section.

23. A: pg 205 – Golden Rectangle and diagonals
A surprising amount of accurate information can be found in the first paragraph on this page. The construction of the rectangle, the description of the diagonals of the rectangles, the diagonals of the squares, and the origin of the spiral are all accurate.

24. A: pg 209 – Fibonacci Sequence
The description of the Fibonacci sequence, as well as the table showing the relationship between the Fibonacci sequence and the Golden Ratio, are accurate.

26. I: pg 211 – Nautilus as a Fibonacci spiral
The Nautilus shell is not a Fibonacci spiral any more than it is a Golden spiral. While it’s growth is logarithmic, it does not even come close to approximating the Fibonacci or Golden spirals.

27. I: pg 211 – Area under the Fibonacci and Golden spirals
The right side of the diagram on p211, labeled “Golden Mean,” does not accurately show the points which would occur on a Golden spiral.

28. I: pg 214 – Classifying spirals as male and female
Mathematically speaking, spirals are always curved lines. The “male spirals” which Drunvalo discusses are simply conjoined line segments. They are not spirals.

29. I: pg 216 – Powers of 2
The set of numbers which Drunvalo calls the “binary sequence” is just the set of all numbers where n is a positive integer. He incorrectly states that three consecutive numbers positively identify the doubling process. There are an infinite number of unique sequences which contain the numbers 2, 4, 8, consecutively. Only one of these is Drunvalo’s “binary sequence.”

While the table of numbers in the “binary sequence” is accurate, Drunvalo goes on to claim that “it takes exactly 46 mitotic cell divisions to reach the cells of the human body.” He apparently forgot to do the math, on this one. If we are starting from a single cell, and we posit that every cell in the system divides at the same moment, after 46 division events we will have 70,368,744,177,664. That number is less than by nearly 30-trillion cells.

30. I: pg 217 – “Binary sequence” as the basis of computing
While Drunvalo’s introduction to the binary number system is mostly correct– if a tad convoluted– he continually forgets to include 0 in his calculations. In his “five chip” example, he says you can represent the numbers 1 to 31. However, you can actually represent the integers 0 to 31. If you add a sixth “chip,” you can represent all the integers from 0 to 63. Another “chip” lets you represent every integer from 0 to 127.

However, his computer with 46 “chips” could NOT represent every number from 1 to 100-trillion, as he claims. It could represent every integer from 0 to 70,368,744,177,663, however.

31. I: pg 219 – Golden Spiral on a Polar Graph
The spiral in the diagram on this page is not actually a Golden spiral. A Golden spiral does not cross the 2nd, 4th, and 8th radials at exactly 120, 240, and 360 degrees. Nor does it intersect the 16th, 32nd, or 64th radials at precisely 120, 240, and 360 degrees.

32. I: pg 221 – Critchlow’s Triangles and Music
Drunvalo does not adequately describe Keith Critchlow’s work, here, nor does he provide a bibliographical reference which would allow me to look up this information, myself. As such, it is not clear to which proportion Critchlow is referring when he mentions the 1/2 (octave), 2/3 (perfect fifth), 4/5 (major third), 8/9 (major tone), and 16/17 (half-tone). However, from my own reproduction of the work, it appears that Critchlow is comparing the line segments which are perpendicular to the triangle’s bisector to half the base of the triangle.

Regardless, Drunvalo clearly does not understand the math involved in the creation of the diagram. He asserts several times that Critchlow measured his drawing to reach these values. This is, however, a preposterous and silly claim. Keith Critchlow is an accomplished architect who has a thorough understanding of geometry. He is not describing measurements, as Drunvalo seems to believe, but proportions. You can arrive at these proportions without ever measuring a single line, simply by understanding geometry.

Then, immediately after impugning Critchlow for supposedly measuring his work, Drunvalo decides to drop the entire diagram onto a polar graph– which is essentially a radial measuring stick. He then draws conclusions based on his measurements from the graph, while claiming that he didn’t measure anything!

That depends entirely upon the definition which you give for the sequence. If you give a definition for the sequence, then knowing one number in the sequence can often be more than enough to calculate the entire sequence. Without a definition for the sequence, no amount of numbers is enough to determine the entire sequence.

For example, let’s say that I give you the first three numbers in a sequence, and these are 1, 2, and 3. To what sequence am I referring? Well, it could be the sequence of natural numbers. Or it could be the sequence of the divisors of 60. Or it could be the sequence of positive integers less than 10. Or it could be the sequence of Fibonacci numbers after the initial 1. Or it could be the sequence of all whole numbers greater than 0 except for 3740. Or it could be any one of an infinite number of other sequences.

Now, let’s say that I define a sequence of numbers such that for all k greater than 1, the kth number in the sequence is equal to one less than twice the k-1st number. Now, let’s say that I tell you the first number in this sequence is 17. You do not need any more numbers from the sequence in order to calculate the whole sequence.

As a sacred geometer who traces the tradition back to roots in
Pythagoras and Plato, I am not a fan of Drunvalo’s writings. However, it is a shame to see you get some of your critiques wrong. Here are some that I have picked out, though your splitting of terminological hairs in other points could be taken on as well.

In point 2 you say the ratio of the vesica piscis is 1 to root 5, it is not. It is 1 to two times root 3. (Root 3 being edge to bisection ratio of the equilateral triangles that fit within the vesica). Please check.

In point 11, the rhombic dodecahedron is an important form that answers your question as to why he makes this distinction.

In point 15 you are certainly wrong about the tetrahedron not fitting point to point inside the cube. 5 cubes will also fit point to point inside a dodecahedron, which is what Drunvalo probably meant in point 14, and yes they are rotated 72 degrees around an axis so that after 5 identical movements they return to the original position.

Im sure if you went through my book ‘Understanding Sacred Geometry and the Flower of Life’ you would find some inaccuracies there, but hopefully it would be more mathematically interesting, and the mistakes would be more subtle than those in Drunvalo’s books 😉 You can (currently) read it fo free online at http://www.UnderstandingSacredGeometry.com

Thanks for taking the time to read and comment! I look forward to reading your book.

Here are some that I have picked out, though your splitting of terminological hairs in other points could be taken on as well.

If there’s anything you’d like to discuss which is either too nitpicky or too detailed to handle in a comment, feel free to email me or write a blog post, and I’ll do my best to respond!

In point 2 you say the ratio of the vesica piscis is 1 to root 5, it is not. It is 1 to two times root 3

Thanks for pointing out the mistake! Honestly not sure where I got root 5 from. That said, I’m don’t think your suggestion is quite right, either. If we take the minor axis of the Vesica Piscis (ie, the radii of our circles) to be 1, then the major axis of the Vesica Piscis (ie, the chord drawn from each point of intersection between the two circles to the other) would measure √3, not √5 or 2√3.

In point 15 you are certainly wrong about the tetrahedron not fitting point to point inside the cube.

Again, thanks for the correction. I had originally been thinking about polyhedral duals, which require each vertex of the inscribed polyhedron to intersect with exactly one face of the circumscribed polyhedron such that all the circumscribed polyhedron’s faces are intersected. If we ignore this latter qualifier, which I neglected to mention in the post, then a tetrahedron could definitely fit the bill.

5 cubes will also fit point to point inside a dodecahedron, which is what Drunvalo probably meant in point 14, and yes they are rotated 72 degrees around an axis so that after 5 identical movements they return to the original position.

It’s certainly possible this is what he meant. The problem is that his description was sorely lacking in any sort of descriptive detail. There are an infinite number of ways to rotate a cube 72 degrees, and almost all of these will NOT produce a regular dodecahedron.

Im sure if you went through my book ‘Understanding Sacred Geometry and the Flower of Life’ you would find some inaccuracies there, but hopefully it would be more mathematically interesting